As relatively-ignored by the Oscars as science fiction films are, it's hard to argue that there is a more influential genre in Hollywood. Ever since Star Wars nominalized the blockbuster, technology has usurped story as the force behind filmmaking. Yes, a good story remains the most important factor in regards to a good movie, but a good story is often not what makes a box office success. You can argue that if you'd like, but all evidence will point against you.
Every time an "event" science fiction film makes its way into theaters, other films (science fiction or not) scramble to take advantage of the technology presented in that event picture. The end result is that motion picture eras can be defined almost as easily by the science fiction film they follow as by the decade they're a part of.
2001: A Space Odyssey (1968)
Stanley Kubrick was infamous for his bucking of Hollywood trends and the irony that he likely kicked off one of its biggest is not lost on anyone (or shouldn't be). There were influential science fiction films before Kubrick and Clarke's 2001: A Space Odyssey, but this was the first film to not only truly emphasize scientific realism in Hollywood science fiction at large, but to also present it as what many might refer to as an art film.
Admittedly, I find 2001: A Space Odyssey a highly overrated narrative mess (the novel, on the other hand, is brilliant), but its influence is obvious. Without it, there would be no Solyaris (1972) or The Man Who Fell to Earth (1976). Indeed, given its own sequel (2010: The Year We Make Contact - 1984) and the underrated American remake of Solyaris (Solaris - 2002), it's easy to argue that 2001: A Space Odyssey remains more influential than the more recent films appearing on this list. 1996's The Arrival and 1997's Contact also directly owe their thematic execution to Kubrick's influence.
Star Wars (1977)
Undoubtedly the most obvious film on the list, this is the one that changed Hollywood from top to bottom. Upon its release and the seemingly miles-long lines of eagerly-awaiting audiences that accompanied it, the film industry started churning out science fiction films (many horribly awful) in droves. Overtly less science fiction and more fairy tale (what many call space opera), Star Wars spanned all of the fantasy genres and its fingerprints can be found on everything from its own sequels to 2006's Eragon.
Specifically regarding science fiction, however, the success of Star Wars convinced Paramount to take a silver screen chance on its beloved franchise, Star Trek, and 20th Century Fox to give Ridley Scott the go-ahead to make a horror film set in space (Star Trek: The Motion Picture and Alien, respectively... both 1979). That the brilliance of The Empire Strikes Back (1980) and the fun of Return of the Jedi (1983) were nearly erased with the despicable Star Wars prequels of the the late 1990s and early 2000s does not lessen the impact of the original movie (and, in fact, proves its resilience).
James Cameron's sequel to Ridley Scott's Alien is, as implied, part of the Star Wars "family tree of influence." And while perhaps not as groundbreaking as Cameron's previous film, The Terminator (1984), Aliens propelled science fiction into the stratosphere (a fact not lost on the Academy Awards, which honored the film with a nomination for Best Actress... a feat unheard of for a science fiction film). Part action, part horror, part commentary, Cameron finally proved with Aliens what stalwarts such as David Cronenberg and Scott were trying to prove with their own (brilliant) films: science fiction could be dark, gritty, and commercially successful.
While some might disagree, Aliens is the true cinematic progenitor of 1991's Terminator 2: Judgment Day (corporate intrigue, strong female protagonist, etc.) and not, as one would imagine, The Terminator. 2000's Pitch Black is even interpreted by some as a quasi-remake of Aliens. 1997's Starship Troopers, too, owes a great deal to Cameron's film (ironically, the novel Starship Troopers heavily influenced Aliens). And lets not forget the plethora of franchise-launching or franchise-renewing projects that appeared because of it: 1987's Predator (which also unfortunately led to 2004's Alien vs. Predator), Alien Nation (1988), a War of the Worlds television series (1988), and even direct and pervasive homage in countless Japanese animated films.
The Matrix (1999)
Strangely, there were rather low expectations for this film to do well at the box office, must less spawn two sequels and a host of animation and graphic novel projects. That it became so influential to Hollywood in general could not have been predicted by anyone. Filmed for a relatively low budget (less than $65 million, where most films of this type are priced well over $100 million), the in-house effects team created some mind-boggling visuals that are still considered top-notch over a decade later. These effects, combined with a comic book-inspired script loaded with (unintentional) relevant philosophy, captured the minds of its audiences while blowing them away with ridiculous (yet appropriate, given the story) action sequences and subdued questions of reality.
Though not entirely relevant to itself, this examination of existence returned the genre to the realm of philosophy. While the tone of such later films as Sunshine (2007) and Moon (2009) doesn't lend to comparison, it would be hard-pressed for one to imagine either of those films being produced without The Matrix. And despite the two Matrix sequels attempting to force the philosophical issue (resulting in far-inferior films), many subsequent films were highly successful at it (2006's The Fountain - despite being a tad too overwrought - and even 2002's 28 Days Later). Essentially, it took an action spectacle to allow the quiet, introspective science fiction film to enjoy a resurgence.
Avatar's place on this list could have very easily been taken by 2007's Transformers (yes, Transformers), but given that film's rush-to-sequel and the enormous amount of press preceding Avatar's release, Transformers had no chance to settle into the subconscious of audiences and filmmakers. There's not a lot to say about Avatar's influence just yet (save for the proliferation of 3D films), but I don't think I'm taking too much of a risk by claiming that, yes, it is the new Star Wars in that filmmaking has been distinctly altered forever.
The story of Avatar is admittedly pedestrian, but out of all of the films on this list, it is the most socially relevant. An intellectual might claim 2001: A Space Odyssey holds that distinction, but Kubrick's film hid its moral behind far too much artistic interpretation that its message went unnoticed by most of its viewers. Avatar, on the other hand, presented its commentary (environmentalism, racism, anti-private military, corporate greed) rather plainly.
It is obvious that Avatar has affected the technical nature of filmmaking for all time to come. It remains to be seen how it affects the presentation of social context within the genre. One can only hope.